Recollement of Deformed Preprojective Algebras and the Calogero–Moser Correspondence

Abstract

The aim of this paper is to clarify the relation between the following objects: (a) rank 1 projective modules (ideals) over the first Weyl algebra A_1; (b) simple modules over deformed preprojective algebras $\Pi_{\lambda}(Q)$ introduced by Crawley-Boevey and Holland; and (c) simple modules over the rational Cherednik algebras $H_{0,c}(S_n)$ associated to symmetric groups. The isomorphism classes of each type of these objects can be parametrized geometrically by the same space (namely, the Calogero-Moser algebraic varieties); however, no natural functors between the corresponding module categories seem to be known. We construct such functors by translating our earlier results on A-infinity modules over A_1 to a more familiar setting of representation theory. In the last section we extend our construction to the case of Kleinian singularities $C^2/\Gamma$, where $\Gamma$ is a finite cyclic subgroup of $SL(2,C)$.